Composite systems and state transformations in topos quantum theory

TL;DR

This paper extends topos quantum theory to include composite systems and state transformations, revealing a bijective correspondence between multipartite probability valuations and positive over pure tensor states, and analyzing the limitations of quantum Markov chains.

Contribution

It introduces a framework for composite systems in topos quantum theory and explores the properties of multipartite probability valuations and their relation to quantum states and transformations.

Findings

Multipartite probability valuations correspond bijectively to positive over pure tensor states.

Quantum Markov chains are trivialized to product valuations due to monogamy constraints.

Incompatibility identified between multipartite valuations and quantum monogamy properties.

Abstract

Topos quantum theory provides representations of quantum states as direct generalizations of the probability distribution, namely probability valuation. In this article, we consider extensions of a known bijective correspondence between quantum states and probability valuations to composite systems and to state transformations. We show that multipartite probability valuations on composite systems have a bijective correspondence to positive over pure tensor states, according to a candidate definition of the composite systems in topos quantum theory. Among the multipartite probability valuations, a special attention is placed to Markov chains which are defined by generalizing classical Markov chains from probability theory. We find an incompatibility between the multipartite probability valuations and a monogamy property of quantum states, which trivializes the Markov chains to product probability valuations. Several observations on the transformations of probability valuations are deduced from the results on multipartite probability valuations, through duality relations between multipartite states and state transformations.